Abstract

The purpose of this paper is to introduce and consider a general implicit iterative process which includes Schu's explicit iterative processes and Sun's implicit iterative processes as special cases for a finite family of generalized asymptotically quasi-nonexpansive mappings. Strong convergence of the purposed iterative process is obtained in the framework of real Banach spaces.

1. Introduction and Preliminaries

Let be a real Banach space and . is said to be uniformly convex if for any there exists such that for any ,

(1.1)

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Let be a nonempty closed and convex subset of a Banach space . Let be a mapping. Denote by the fixed point set of .

Recall that is said to be nonexpansive if

(1.2)

is said to bequasi-nonexpansive if and

(1.3)

A nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive; however, the inverse may be not true. See the following example [1].

Example 1.1.

Let and define a mapping by by

(1.4)

Then is quasi-nonexpansive but not nonexpansive.

is said to beasymptotically nonexpansive if there exists a positive sequence with as such that

(1.5)

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence . The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972. It is known that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive mapping on has a fixed point. Further, the set of fixed points of is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

is said to beasymptotically quasi-nonexpansive if , and there exists a positive sequence with as such that

(1.6)

is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

(1.7)

Putting , we see that as . Then (1.7) is reduced to the following:

(1.8)

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [3] (see also Bruck et al. [4]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if is a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point; see [5] more details.

is said to beasymptotically quasi-nonexpansive in the intermediate sense if it is continuous, , and the following inequality holds:

(1.9)

Putting , we see that as . Then (1.9) is reduced to the following:

(1.10)

is said to be generalized asymptotically nonexpansive if there exist two positive sequences with and with as such that

(1.11)

It is easy to see that the class of generalized asymptotically nonexpansive includes the class of asymptotically nonexpansive as a special case.

is said to begeneralized asymptotically quasi-nonexpansive if , and there exist two positive sequences with and with as such that

(1.12)

The class of generalized asymptotically quasi-nonexpansive was considered by Shahzad and Zegeye [6]; see [6, 7] for more details.

Recall that the modified Mann iteration which was introduced by Schu [8] generates a sequence in the following manner:

(1.13)

where is a sequence in the interval and : is an asymptotically nonexpansive mapping.

Let be a uniformly convex Banach space, closed bounded and convex, and asymptotically nonexpansive with sequence for which and is bounded away. Let be a sequence generated in (1.13). Then .

Theorem Schu 2.

Let be a uniformly convex Banach space, closed bounded and convex, and asymptotically nonexpansive with sequence for which and is bounded away. Let be a sequence generated in (1.13). Suppose that is compact for some positive integer . Then the sequence converges strongly to some fixed point of .

Theorem Schu 3.

Let be a uniformly convex Banach space, closed bounded and convex, and asymptotically nonexpansive with sequence for which and is bounded away. Let be a sequence generated in (1.13). Suppose that there exists a nonempty compact and convex subset of and such that

(1.14)

Then the sequence converges strongly to some fixed point of .

In 2007, Shahzad and Zegeye [6] considered the following implicit iterative process for a finite family of generalized asymptotically quasi-nonexpansive mappings :

(1.15)

where is the initial value and is a sequence . Since for each , it can be written as , where , is a positive integer, and as . Hence the above table can be rewritten in the following compact form:

(1.16)

We remark that the implicit iterative process (1.16) was first considered by Sun [9]; see [9] for more details.

Let be a real uniformly convex Banach space and be a nonempty closed convex subset of . Let , where , be uniformly Lipschitz, generalized asymptotically quasi-nonexpansive self-mappings of with , such that and for all . Suppose that and there exists one member in which is either semicompact or satisfies condition . Let for some . From arbitrary , define the sequence by (1.16). Then converges strongly to a common fixed point of the mappings .

Theorem SZ 2.

Let be a real uniformly convex Banach space and a nonemptyclosed convex subset of . Let , where , be generalized asymptotically quasi-nonexpansive self-mappings of with , such that and for all . Suppose that is closed. Let for some . From arbitrary , define the sequence by (1.16). Then converges strongly to a common fixed point of the mappings if and only if .

In this paper, motivated by the above results, we consider the following implicit iterative process for two finite families of generalized asymptotically quasi-nonexpansive mappings and :

(1.17)

where is the initial value, is a bounded sequence in , and , , , and are sequences such that for each . Since for each , it can be written as , where , is a positive integer and as . Hence the above table can be rewritten in the following compact form:

(1.18)

We remark that our implicit iterative process (1.18) which includes the explicit iterative process (1.13) and the implicit iterative process (1.16) as special cases is general.

If , where denotes the identity mapping, for each , then the implicit iterative process (1.18) is reduced to the following implicit iterative process:

(1.19)

If , where denotes the identity mapping, for each , then the implicit iterative process (1.18) is reduced to the following explicit iterative process:

(1.20)

The purpose of this paper is to study the convergence of the implicit iteration process (1.18) for two finite families of generalized asymptotically quasi-nonexpansive mappings. Strong convergence theorems are obtained in the framework of real Banach spaces. The results presented in this paper improve and extend the corresponding results in Shahzad and Zegeye [6], Sun [9], Chang et al. [10], Chidume and Shahzad [11], Guo and Cho [12], Kim et al. [13], Qin et al. [14], Thianwan and Suantai [15], Xu and Ori [16], and Zhou and Chang [17].

In order to prove our main results, we also need the following lemmas.

Let be a real uniformly convex Banach space, a positive number, and a closed ball of . Then there exists a continuous, strictly increasing, and convex function with such that

(1.22)

for all and such that .

2. Main Results

Lemma 2.1.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each and a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

Then

(2.1)

Proof.

First, we show that the sequence generated in (1.18) is well defined. For each , define a mapping as follows:

(2.2)

Notice that

(2.3)

From the restriction (a), we see that is a contraction for each . From Banach contraction mapping principle, we can prove that the sequence generated in (1.18) is well defined.

Fixing , we see that

(2.4)

Notice that . We see from the restrictions (a) and (b) that there exists a positive integer such that

(2.5)

where . It follows from (2.4) that

(2.6)

where is an appropriate constant such that . In view of the restrictions (a) and (b), we obtain from Lemma 1.2 that exists. It follows that the sequence is bounded. In view of Lemma 1.3, we see that

(2.7)

where and are appropriate constants such that and . This implies that

(2.8)

In view of the restrictions (a) and (b), we obtain that

(2.9)

Since is a continuous, strictly increasing, and convex function with , we obtain that

(2.10)

Next, we show that

(2.11)

From Lemma 1.3, we also see that

(2.12)

This implies that

(2.13)

In view of the restrictions (a) and (b), we obtain that

(2.14)

Since is a continuous, strictly increasing, and convex function with , we obtain that (2.11) holds. Notice that

(2.15)

In view of (2.10) and (2.11), we see from the restriction (b) that

(2.16)

which implies that

(2.17)

Since for any positive integer , it can be written as , where , observe that

(2.18)

Since for each , (mod ), on the other hand, we obtain from that . That is,

(2.19)

Notice that

(2.20)

Substituting (2.20) into (2.18), we arrive at

(2.21)

In view of (2.11) and (2.17), we obtain that

(2.22)

Notice that

(2.23)

It follows from (2.16) and (2.22) that

(2.24)

Notice that

(2.25)

From (2.17) and (2.24), we arrive at

(2.26)

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

(2.27)

Letting , we have

(2.28)

In view of (2.10) and (2.16), we see that

(2.29)

Observe that

(2.30)

In view of

(2.31)

we arrive at

(2.32)

In view of (2.10), (2.17), and (2.29), we obtain that

(2.33)

Notice that

(2.34)

From (2.16) and (2.33), we see that

(2.35)

On the other hand, we have

(2.36)

It follows from (2.17) and (2.35) that

(2.37)

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

(2.38)

This completes the proof.

Recall that a mapping is said to be semicompact if for any bounded sequence in such that as , then there exists a subsequence such that .

Next, we give strong convergence theorems with the help of the semicompactness.

Theorem 2.2.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

If one of or one of is semicompact, then the sequence converges strongly to some point in .

Proof.

Without loss of generality, we may assume that is semicompact. From (2.38), we see that there exits a subsequence of converging strongly to . For each , we get that

(2.39)

Since is Lipshcitz continuous, we obtain from (2.38) that . Notice that

(2.40)

Since is Lipshcitz continuous, we obtain from (2.27) that . This means that . In view of Lemma 2.1, we obtain that exists. Therefore, we can obtain the desired conclusion immediately.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.3.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

(b).

If one of is semicompact, then the sequence converges strongly to some point in .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.4.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , for all ;

(b).

If one of is semicompact, then the sequence converges strongly to some point in .

In 2005, Chidume and Shahzad [11] introduced the following conception. Recall that a family with is said to satisfy Condition on if there is a nondecreasing function with and for all such that for all

(2.41)

Based on Condition , we introduced the following conception for two finite families of mappings. Recall that two families and with are said to satisfy Condition on if there is a nondecreasing function with and for all such that for all

(2.42)

Next, we give strong convergence theorems with the help of Condition .

Theorem 2.5.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

If and satisfy Condition , then the sequence converges strongly to some point in .

Proof.

In view of Condition , we obtain from (2.27) and (2.38) that , which implies . Next, we show that the sequence is Cauchy. In view of (2.6), for any positive integers , where , we see that

(2.43)

where . It follows that

(2.44)

It follows that is a Cauchy sequence in and so converges strongly to some . Since and are Lipschitz for each , we see that is closed. This in turn implies that . This completes the proof.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.6.

Let be a real uniformly convex uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and where . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

(b).

If satisfies Condition , then the sequence converges strongly to some point in .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.7.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , and , for all ;

(b).

If satisfies Condition , then the sequence converges strongly to some point in .

Finally, we give a strong convergence theorem criterion.

Theorem 2.8.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each , and let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , where and and , where and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.18). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , where , for all ;

(b).

Then converges strongly to some point in if and only if .

Proof.

The necessity is obvious. We only show the sufficiency. Assume that

(2.45)

For each , we see that

(2.46)

Notice that . We see from the restrictions (a) and (b) that there exists a positive integer such that

(2.47)

where . Notice that the sequence is bounded. It follows from (2.46) that

(2.48)

where is an appropriate constant such that . In view of the restrictions (a) and (b), we obtain from Lemma 1.2 that exists. This implies that

(2.49)

In view of Theorem 2.5, we can conclude the desired conclusion easily.

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.9.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , and where . Let , , and be sequences in such that for each . Let be a sequence generated in (1.19). Assume that the following restrictions are satisfied:

(a)there exist constants such that and , where , for all ;

(b).

Then converges strongly to some point in if and only if .

If , where denotes the identity mapping, for each , then Theorem 2.2 is reduced to the following.

Corollary 2.10.

Let be a real Banach space and a nonempty closed convex subset of . Let be a uniformly -Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences and such that and for each . Assume that is nonempty. Let be a bounded sequence in , , and . Let , , , and be sequences in such that for each . Let be a sequence generated in (1.20). Assume that the following restrictions are satisfied:

(a)there exist constants such that , , and , for all ;

(b).

Then converges strongly to some point in if and only if .

Authors’ Affiliations

(1)

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power

(2)

Department of Mathematics and the RINS, Gyeongsang National University

(3)

Department of Mathematical Sciences, Florida Institute of Technology

(4)

Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals

Saejung S, Suantai S, Yotkaew P: a note on common fixed point of multistep Noor iteration with errors for a finite family of generalized asymptotically quasi-nonexpansive mapping.Abstract and Applied Analysis 2009, 2009:-9.Google Scholar

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